Vol 96, No 2 (2017)

The Dirichlet problem for a one-dimensional (with respect to x) second-order parabolic system with Dini continuous coefficients is considered in an x-semibounded domain with a nonsmooth lateral boundary from the Dini–Hölder class. The classical solvability of the problem is proved by applying the method of boundary integral equations. The only condition imposed on the right-hand side of the boundary condition is that it has a continuous derivative of order 1/2 vanishing at t = 0. The smoothness of the solution is studied.

We deduce a system of ODEs describing the behavior of critical points and poles of a smooth one-parametric family of rational functions uniformizing a given family of ramified coverings of the Riemann sphere.

The properties of (q₁, q₂)-quasimetric spaces are examined. Multivalued covering mappings between (q₁, q₂)-quasimetric spaces are investigated. Given two multivalued mappings between (q₁, q₂)-quasimetric spaces such that one of them is covering and the other satisfies the Lipschitz condition, sufficient conditions for these mappings to have a coincidence point are obtained. A theorem on the stability of coincidence points with respect to small perturbations in the considered mappings is proved.

A functional Hamilton–Jacobi equation with covariant derivatives which corresponds to neutral-type dynamical systems is obtained. The definition of a minimax solution of this equation is given. Conditions under which such a solution exists and is unique and well defined are found.

Let φ(F) be the property of containing (as a subgraph) an isomorphic copy of a graph F. It is easy to show that this property cannot be defined in a first-order language by a sentence with a quantifier depth (or variable width) strictly less than the number of vertices in F. Nevertheless, such a definition exists in some classes of graphs. Three classes of graphs are considered: connected graphs with a large number of vertices, graphs with large treewidth, and graphs with high connectivity.

There are two versions of weighted vector algorithms for the statistical modeling of polarized radiative transfer: a “standard” one, which is convenient for parametric analysis of results, and an “adaptive” one, which ensures finite variances of estimates. The application of the adaptive algorithm is complicated by the necessity of modeling the previously unknown transition density. An optimal version of the elimination algorithm used in this case is presented in this paper. A new combined algorithm with a finite variance and an algorithm with a mixed transition density are constructed. The comparative efficiency of the latter is numerically studied as applied to radiative transfer with a molecular scattering matrix.

Conditions for commuting a Toeplitz matrix and a Hankel matrix were obtained relatively recently (in 2015). The solution to the problem of describing all anti-commuting pairs (T, H), where T is a Toeplitz matrix and H is a Hankel matrix, is sketched below.

Results are obtained that substantially strengthen a previously known bound for the chromatic number of a random subgraph of the Kneser graph.

For the numerical solution of nonstationary quasilinear hyperbolic equations, a family of symmetric semidiscrete bicompact schemes based on collocation polynomials is constructed in the one- and multidimensional cases. A dispersion analysis of a semidiscrete bicompact scheme of six-order accuracy in space is performed. It is proved that the dispersion properties of the scheme are preserved on highly nonuniform spatial grids. It is shown that the phase error of the sixth-order bicompact scheme does not exceed 0.2% in the entire range of dimensionless wave numbers. A numerical example is presented that demonstrates the ability of the bicompact scheme to adequately simulate wave propagation on coarse grids at long times.

A learning-based classification problem with a large number of classes is considered. The error-correcting-output-codes (ЕСОС) scheme is optimized. An initial binary matrix is formed at random so that the number of its rows is equal to the number of classes and each column corresponds to the union of several classes in two macroclasses. In the ЕСОС approach, a binary classification problem is solved for every object to be recognized and for every union. The object is assigned to the class with the nearest code row. A generalization of the ЕСОС approach is presented in which a discrete optimization problem is solved to find optimal unions, probabilities of correct classification are used in dichotomy problems, and the degree of dichotomy informativeness is taken into account. If the solution algorithms for the dichotomy problems are correct, the recognition algorithm for the original problem is correct as well.

An extended version of the principle of empirical risk minimization is proposed. It is based on the application of averaging aggregation functions, rather than arithmetic means, to compute empirical risk. This is justified if the distribution of losses has outliers or is substantially distorted, which results in that the risk estimate becomes biased from the very beginning. In this case, for optimizing parameters, a robust estimate of the mean risk should be used. Such estimates can be constructed by using averaging aggregation functions, which are the solutions of the problem of minimizing the function of penalty for deviation from the mean value. An iterative reweighting scheme for numerically solving the problem of empirical risk minimization is proposed. Illustrative examples of the construction of a robust procedure for estimating parameters in the linear regression problem and in the problem of linearly separating two classes based on the application of an averaging mean function, which replaces the α-quantile, are given.

A complete proof that algorithms proposed by the authors solve the problem of minimum-cost transformation of a graph into another graph is given. The problem is solved both by a direct algorithm of linear complexity and by a reduction to quadratic integer linear programming.

A Steklov-type problem with rapidly alternating Dirichlet and Steklov boundary conditions in a bounded n-dimensional domain in considered. The regions on which the Steklov condition is given have diameter of order ε, and the distance between them is larger than or equal to 2ε. It is proved that, as the small parameter tends to zero, the eigenvalues of this problem degenerate, i.e., tend to infinity. It is also proved that the rate of increase to infinity is larger than or equal to |ln ε|^δ, δ ∈ (0;2 − 2/n) as ε, tends to zero.

Seismic waves propagating in a fractured geological medium are numerically simulated. Their dynamic behavior is described using a linear elastic model with an explicit description of all crack boundaries (a contact discontinuity problem is solved). An algorithm for seismic imaging of the fractured medium is proposed. A distinctive feature of this approach is the use of an initially fractured background model. The forward and adjoint wave fields are numerically computed by applying the grid-characteristic method on hexahedral meshes.

A structure and implementation principles of a photon computer are proposed. Its functioning is based on effects of the interaction between coherent light wave systems generated by a laser source. The performance of photon computers, consumed energy, and physical sizes are estimated. These estimates show possible advantages of photon computers over electronic ones.

A new method for calculating space vehicle (SV) attitude controls ensuring their effective implementation by a system of collinear pairs of single-gimbal forced unrestrained gyros (gyrodynes) has been proposed. The novelty of the method consists in a virtual kinematic configuration of the gyro system, i.e., the precession of gyro units in the collinear pairs of gyrodynes is coupled in a nonmechanical manner. In addition, the angular momentum of the system as a state variable for describing the dynamics of the SV permanent rotation was used for the first time at the stage of computing controls performed nonstop by gyrodynes. In the general formulation, when the desired final state of the SV is arbitrary, the SV attitude control problem can be reduced to a sequence of permanent rotations. The performance of the method is demonstrated as applied to the calculation of program gyrodyne controls with a permanent reduction in the SV angular velocity around its center of mass with a nonzero SV angular momentum after its discharge.

Sufficient conditions for an endpoint cost criterion in a controllable stochastic diffusion system to be constant with probability 1 (i.e., for weak invariance condition) and sufficient conditions for absolute invariance, i.e., the independence of an endpoint cost criterion on the realization of the random process and the initial data, are obtained.

An approach based on optimization methods is developed for visualizing multidimensional frontiers in nonconvex FDH models. This approach has earlier been used for frontier visualization in DEA models. Such an approach facilitates the computation of various scale characteristics in FDH models.